Geometric Average Returns


BUSI 721, Fall 2022
JGSB, Rice University

Kerry Back

Tesla


  • Tesla went down 50% between Nov 2021 and May 2022.
  • It then went up 50% between May 2022 and Aug 2022.
  • Were Tesla shareholders back to even?

  • For each $100 of Tesla stock, shareholders experienced 100 → 50

  • and then 50 → 75.

  • They lost 25%, even though the average return was zero.

  • So, lose 50% and make 50% → lose 25%. Suppose you

    • make 50% and then lose 50%?

    • lose 50% and then make 100%?

    • make 100% and then lose 50%?

Geometric Average Return

Given returns \(r_1, \cdots, r_n,\)
the geometric average return is the number \(r\) such that

\[(1+r)^{n}=(1+r_1)\cdots(1+r_{n})\]

So earning \(r\) each period produces the same accumulation as the actual returns \(r_1, \cdots, r_n.\)


We solve for it as

\[r=[(1+r_1)\cdots(1+r_n)]^{1/n}-1\]

The geometric average return is always less than the arithmetic average return.



Examples

  • make 50% and lose 50% → geometric average is

\[\sqrt{1.5 \times 0.5}-1=-0.134\]

  • make 100% and lose 50% → geometric average is

\[\sqrt{2 \times 0.5}-1=0\]

Averaging continuously compounded returns

  • Recall that compounding continuously at rate \(z\) means $1 will grow to \(e^z\) in one year.
  • If the actual return is \(r\), then the corresponding continuously compounded return is \(\log (1+r)\), because compounding continuously at rate \(\log (1+r)\) means $1 will grow to \(e^{\log (1+r)} = 1+r\).
  • Make 50% and lose 50% \(\Rightarrow\) continuously compounded returns are \(\log(1.5)\) and \(\log(0.5)\) and average \(<0\).
  • Make 100% and lose 50% \(\Rightarrow\) continuously compounded returns are \(\log(2)\) and \(\log(0.5)\) and average \(=0\).